STUDENT MATHEMAT ICAL L IBRARY Volume 80

Game Theory

A Playful Introduction

Matt DeVos

Deborah A. Kent

Game Theory A Playful Introduction

Game Theory A Playful Introduction

Matt DeVosDeborah A. Kent

STUDENT MATHEMAT ICAL L IBRARYVolume 80

Providence, Rhode Island

https://doi.org/10.1090//stml/080

Editorial BoardSatyan L. DevadossErica Flapan

John Stillwell (Chair)Serge Tabachnikov

2010Mathematics Subject Classification. Primary 91-01, 91A46, 91A06, 91B06.

For additional information and updates on this book, visitwww.ams.org/bookpages/stml-80

Library of Congress Cataloging-in-Publication DataNames: DeVos, Matthew Jared, 1974- | Kent, Deborah A., 1978-Title: Game theory : a playful introduction / Matthew DeVos, Deborah A. Kent.Description: Providence, Rhode Island : American Mathematical Society, [2016] | Series:

Student mathematical library ; volume 80 | Includes bibliographical references andindex.

Identifiers: LCCN 2016035452 | ISBN 9781470422103 (alk. paper)Subjects: LCSH: Game theory–Textbooks. | Combinatorial analysis–Textbooks. | AMS:

Game theory, economics, social and behavioral sciences – Instructional exposition(textbooks, tutorial papers, etc.). msc | Game theory, economics, social and behavioralsciences – Game theory – Combinatorial games. msc |Game theory, economics, socialand behavioral sciences – Game theory – n-person games, n > 2. msc | Game the-ory, economics, social and behavioral sciences – Mathematical economics – Decisiontheory. msc

Classification: LCC QA269 .D45 2016 | DDC 519.3–dc23 LC record available athttps://lccn.loc.gov/2016035452

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Dedicated to my family, MD.For Mom, in memory with love, DK.

Contents

Preface xi

Chapter 1. Combinatorial Games 1§1.1. Game Trees 3§1.2. Zermelo’s Theorem 9§1.3. Strategy 14Exercises 19

Chapter 2. Normal-Play Games 25§2.1. Positions and Their Types 27§2.2. Sums of Positions 30§2.3. Equivalence 36Exercises 41

Chapter 3. Impartial Games 45§3.1. Nim 46§3.2. The Sprague-Grundy Theorem 52§3.3. Applying the MEX Principle 54Exercises 59

vii

viii Contents

Chapter 4. Hackenbush and Partizan Games 63§4.1. Hackenbush 64§4.2. Dyadic Numbers and Positions 71§4.3. The Simplicity Principle 77Exercises 83

Chapter 5. Zero-SumMatrix Games 89§5.1. Dominance 91§5.2. Mixed Strategies 95§5.3. Von Neumann Solutions 100Exercises 104

Chapter 6. Von Neumann’s Minimax Theorem 111§6.1. Equating the Opponent’s Results 113§6.2. Two-Dimensional Games 118§6.3. Proof of the Minimax Theorem 123Exercises 128

Chapter 7. General Games 133§7.1. Utility 135§7.2. Matrix Games 139§7.3. Game Trees 145§7.4. Trees vs. Matrices 150Exercises 155

Chapter 8. Nash Equilibrium and Applications 161§8.1. Nash Equilibrium 162§8.2. Evolutionary Biology 169§8.3. Cournot Duopoly 176Exercises 182

Chapter 9. Nash’s Equilibrium Theorem 187§9.1. Sperner’s Lemma 189§9.2. Brouwer’s Fixed Point Theorem 192

Contents ix

§9.3. Strategy Spaces 198§9.4. Nash Flow and the Proof 202Exercises 208

Chapter 10. Cooperation 213§10.1. The Negotiation Set 214§10.2. Nash Arbitration 221§10.3. Repeated Games and the Folk Theorem 228Exercises 238

Chapter 11. 𝑛-Player Games 245§11.1. Matrix Games 247§11.2. Coalitions 251§11.3. Shapley Value 260Exercises 270

Chapter 12. Preferences and Society 275§12.1. Fair Division 277§12.2. Stable Marriages 285§12.3. Arrow’s Impossibility Theorem 290Exercises 298

Appendix A. On Games and Numbers 301

Appendix B. Linear Programming 309Basic Theory 310A Connection to Game Theory 313LP Duality 317

Appendix C. Nash Equilibrium in High Dimensions 323

Game Boards 331

Bibliography 335

Index of Games 339

Index 341

Preface

The story of this book began in 2002whenMatt, then a postdoc at Prince-ton University, was given the opportunity to teach an undergraduateclass in game theory. Thanks largely to the 2001 release of a Hollywoodmovie on the life of the famous Princeton mathematician and (classical)game theorist JohnNash, this course attracted a large and highly diverseaudience. Princeton’s mathematics department featured not only Nash,but also John Conway, the father of modern combinatorial game theory.So it seemed only natural to blend the two sides of game theory, combi-natorial and classical, into one (rather ambitious) class. The varied back-grounds of the students and the lack of a suitable textbook made for anextremely challenging teaching assignment (that sometimes went awry).However, the simple fun of playing games, the richmathematical beautyof game theory, and its significant real-world connections still made foran amazing class.

Deborah adopted a variant of this material a few years later and fur-ther developed it for a general undergraduate audience. Over the ensu-ing years, Deborah andMatt have both taught numerous incarnations ofthis course at various universities. Through exchange and collaboration,the material has undergone a thorough evolution, and this textbook rep-resents the culmination of our process. We hope it will provide an intro-ductory course in mathematical game theory that you will find inviting,entertaining, mathematically interesting, and meaningful.

xi

xii Preface

Combinatorial game theory is the study of games like Chess andCheckers in which two opponents alternate moves, each trying to winthe game. This part of game theory focuses on deterministic games withfull information and is thus highly amenable to recursive analysis. Com-binatorial game theory traces its roots to Charles Bouton’s theory of thegame Nim and a classification theorem attributed independently toRoland Sprague and Patrick Grundy. The 1982 publication of the classicWinning Ways for Your Mathematical Plays by Elwyn Berlekamp, JohnConway, and Richard Guy laid a modern foundation for the subject—now a thriving branch of combinatorics. In contrast, classical game the-ory is an aspect of appliedmathematics frequently taught in departmentsof economics. Classical game theory is the study of strategic decision-making in situations with two or more players, each of whom may af-fect the outcome. John von Neumann and Oskar Morgenstern are com-monly credited with the foundation of classical game theory in theirgroundbreakingworkTheory ofGames andEconomicBehavior publishedin 1944. This treatise established a broad mathematical framework forreasoning about rational decision-making in a wide variety of contextsand it launched a new branch of academic study. Although there havebeen many significant developments in this theory, John Nash meritsmention for his mathematical contributions, most notably the NashEquilibrium Theorem.

Traditionally, the classical and combinatorial sides of game theoryare separated in the classroom. A strong theme of strategic thinkingnonetheless connects them and we have found the combination to re-sult in a rich and engaging class. The great fun we have had teachingthis broad mathematical tour through game theory undergirded our de-cision to write this book. From the very beginning of this project, ourgoal has been to give an honest introduction to themathematics of gametheory (both combinatorial and classical) that is accessible to an early un-dergraduate student. Over the years, we have developed an approach toteaching combinatorial game theory that avoids some of the set-theoreticcomplexities found in advanced treatments yet still holds true to the sub-ject. As a result, we achieve the two cornerstones of the Sprague-GrundyTheoremand the Simplicity Principle in an efficient and student-friendly

Preface xiii

manner. The classical game theory portion of the book contains numer-ous carefully sculpted and easy-to-follow proofs to establish the theoret-ical core of the subject (including the Minimax Theorem, Nash arbitra-tion, Shapley Value, and Arrow’s Paradox). Most significantly, Chapter9 is entirely devoted to an extremely gentle proof of Nash’s EquilibriumTheorem. For the sake of concreteness, the chapter focuses on 2 × 2matrices, but each argument generalizes and Appendix C contains fulldetails. Sperner’s Lemma appears in this chapter as the first step of ourproof and we offer an intuitive exposition of this lemma by treating it asa game of solitaire. More broadly, Sperner’s Lemma provides a touch-stone through other chapters. In addition to using it to prove Nash’sEquilibrium Theorem, we also call on it to show that the combinatorialgame Hex cannot end in a draw. Later still, Sperner’s Lemma allows usto construct an envy-free division of cake.

Beyond including both combinatorial and classical theory, we havesought to provide a broad overview of (both sides of) the subject. Withinthe world of combinatorial game theory, we begin at a very high level ofgenerality with game trees and Zermelo’s Theorem—concepts that ap-ply to Chess, Checkers, and many other 2-player games. We also intro-duce some widely applicable ideas such as symmetry and strategy steal-ing before specializing in normal-play games to develop the heart of thetheory. On the classical side, in addition to the essential mathematicalconcepts, we tour a variety of exciting supplementary topics includingthe Folk Theorem, cake cutting, and stable marriages. Furthermore, wehave devoted considerable effort to connecting the theory with applica-tions. Chapter 7 focuses on the modeling capability of a game-theoreticframework in the context of sports, biology, business, politics, andmore!

One of our primary goals in this book is to enhance the mathemat-ical development of our student readers. Indeed, we aim to take advan-tage of the naturally stimulating subject of game theory to teach mathe-matics. We have found that blending combinatorial and classical gametheory has great pedagogical advantages. Beginning with combinatorialgames means that student pairs are playing and recursively analyzinggames right from the start. These games are not only fun to play, butthey provide a perfect environment for working with game trees, prov-ing theorems by induction, and starting to think strategically. This part

xiv Preface

of the book features numerous rich examples of proofs by induction andalso a number of interesting proofs by contradiction. Turning to clas-sical game theory, we encounter basic probability, linear algebra, andconvexity in our study of zero-sum matrix games. Our later chapters ongeneral games continue to emphasize probability and geometric meth-ods but also introduce questions of modeling as well as plentiful appli-cations. The proof of Nash’s Equilibrium Theorem involves a nice blendof combinatorial and continuous mathematics in addition to a taste oftopology. Whenever a significant newmathematical concept is required,we pause to introduce it; accordingly, this book contains elementary in-troductions to proofs by induction, proofs by contradiction, probability,and convexity.

We have constructed this textbook for a one-semester undergradu-ate course aimed at students who have already taken courses in differen-tial calculus and linear algebra. However, we have found this materialadaptable to a variety of situations and a range of audiences. In particu-lar, most of the book does not directly call upon either calculus or linearalgebra and is thus suitable for students who lack these prerequisites buthave a similar level of sophistication. Indeed, calculus is used very rarely,and for a capable student without linear algebra, only the proofs of theMinimax and Equilibrium Theorems would be out of reach after a quickintroduction to matrix multiplication. The complete book is likely morematerial than can be comfortably covered in a standard undergraduatesemester 3-credit course. To allow the instructor considerable flexibilityin content choices, we have limited dependencies between the chapters(see the diagram on page xv). These limited dependencies also allow forportions of this book to be used in other contexts. For instance, the firstfour chapters on combinatorial games provide an appealing theme for anintroductory proofs course, Chapters 5 and 6 on zero-summatrix gamestogether with Appendix B on linear programming make a nice additionto a linear algebra course, and all three sections in Chapter 12 can betaught independently.

Further to assist the instructor, each chapter ends with a generoussupply of exercises. We have sought to include problems at a variety oflevels from basic skills all the way up to challenging proofs, with espe-

Preface xv

n-Player

2-Player

Cooperative

Individual

General

Zero-Sum

Classical

Combinatorial

Preferences and

Society1211 n-Player Games

Cooperation10

Nash's Equilibrium

Theorem9

Nash Equilibrium

and Applications8

General Games7

Von Neumann's

Minimax Theorem6

Zero-Sum Matrix

Games5

Combinatorial

Games1

Normal-Play

Games2

Impartial

Games3

Hackenbush and

Partizan Games4

9:1

6:2

Figure 0.1. Implication Diagram

xvi Preface

cially difficult exercises marked with the symbol *. References to exer-cises in the same chapter are by exercise number, while those to exercisesin another chapter also include the chapter number. In addition, gameboards and further supplementary material can be found online at

www.ams.org/bookpages/stml-80.This book owes its existence to the many amazing teachers from

whom we have been fortunate to learn. Matt’s genesis as a combinato-rialist is thanks to his incomparable PhD supervisor, Paul Seymour. Healso benefited from an inspiring introduction to combinatorial gamesfrom John Conway and a detailed initiation to the mathematics of clas-sical game theory under the guidance of Hale Trotter. Deborah deeplyappreciates her inimitable dissertation advisor, Karen Parshall, who in-troduced her to the joys and labors of academic mathematics. She alsothanks Tom Archibald for his generous support of this and her otherpostdoctoral projects. We are so grateful to many of our friends andcolleagues who have influenced the development of this book either di-rectly or indirectly: Derek Smith, DragoBokal, Francis Su, ClaudeTardif,and Dave Muraki top this list, but there are countless others. We owe adebt of gratitude to the universities that made it possible for us to teachversions of this class and to the many students who helped to shape thismaterial with their questions, comments, and corrections. We wouldalso like to thank InaMette, Arlene O’Sean, Courtney Rose, and the restof the editorial staff at the AMS whose careful work on our manuscriptdramatically improved the final product. Finally, we thank our friendsand especially our families for their amazing support throughout the ex-tensive process of creating this book. Although it has taken far moreeffort and energy than we could ever have foreseen, writing this bookhas been a labor of love for us. We hope you will enjoy it, too!

Game Boards

Chop

Chomp

These game boards are available online at www.ams.org/bookpages/stml-80.

331

Louise(bLack)

toplayfirst

Richard(gRay)to

playfirst

21

3

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ology 299 (2012), 21–24.6. Emmanual N. Barron, Game theory: An introduction, 2nd ed., JohnWiley &

Sons, 2013.7. Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy, Winning ways

for your mathematical plays. vol. 1, 2nd ed., A K Peters, Ltd., 2001.8. Ken Binmore, Game theory: A very short introduction, Oxford University

Press, 2007.9. , Playing for real, Oxford University Press, 2007.10. André Bouchet, On the Sperner lemma and some colorings of graphs, J. Com-

binatorial Theory Ser. B 14 (1973), 157–162.11. Charles L. Bouton, Nim, a game with a complete mathematical theory, Ann.

of Math. (2) 3 (1901/02), no. 1-4, 35–39.12. John H. Conway, On numbers and games, 2nd ed., A K Peters, Ltd., 2001.13. Antoine Augustin Cournot and Irving Fisher, Researches into the mathemat-

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17. Avinash K. Dixit, Susan Skeath, and David Reiley,Games of strategy, Norton,1999.

18. Thomas S. Ferguson, Game theory, 2nd ed., 2014.19. Len Fisher, Rock, paper, scissors: Game theory in everyday life, Basic Books,

2008.20. Drew Fudenberg and Eric Maskin, The folk theorem in repeated games with

discounting or with incomplete information, Econometrica 54 (1986), no. 3,533–554.

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riage, Amer. Math. Monthly 120 (2013), no. 5, 386–391.25. RickGillman andDavidHousman,Models of conflict and cooperation, Amer-

ican Mathematical Society, 2009.26. Patrick M. Grundy,Mathematics and games, Eureka 2 (1939), 6–8.27. William D. Hamilton and Robert Axelrod, The evolution of cooperation, Sci-

ence 211 (1981), no. 27, 1390–1396.28. MichaelHenle,Acombinatorial introduction to topology, Dover Publications,

Inc., 1994.29. John F. Banzhaf III,Weighted voting doesn’t work: A mathematical analysis,

Rutgers Law Rev. 19 (1964), 317.30. Ehud Kalai and Meir Smorodinsky, Other solutions to Nash’s bargaining

problem, Econometrica 43 (1975), 513–518.31. Donald E. Knuth, Surreal numbers, Addison-Wesley Publishing Co., 1974.32. Alexander Mehlmann, The game’s afoot! Game theory in myth and paradox,

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& Sons, Ltd., 1986.51. Lloyd S. Shapley, A value for n-person games, Ann. Math. Stud. 28 (1953),

307–317.52. , Game theory, Notes for Mathematics 147 at UCLA, 1991.53. Lloyd S. Shapley andMartin Shubik,Amethod for evaluating the distribution

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Index of Games

The bold page numbers in index entries are the pages on whichthe term is defined.

2/3 of the Average Game, 250, 2513D Chop, 61

AKQ, 157

Chomp, 2, 17, 55Chop, 1, 15, 54, 56, 57Coin Jack, 157Coin Poker, 149, 150, 151Coin Toss, 147Colonel Blotto, 104Common Side-Blotched Lizard, 172,

176Competing Pubs, 142Coordination Game, 140, 173, 174, 176Cut-Cake, 26, 81

Dating Dilemma, 140, 145, 205, 218,220

Divide the Dollar, 256, 257–260Domineering, 34, 81, 83

Empty and Divide, 59, 61Euclid’s Game, 44Even-Nim, 60

General Volunteering Dilemma, 248

Hackenbush, 63, 64, 65–71, 74Hawk vs. Dove, 170, 171–173, 176

Heap, 61Hex, 2, 17, 18, 209, 210

Infinite Nim, 61Investing Dilemma, 249

Kayles, 23

Moving Knife, 278

Newcomb’s Paradox, 160Nim, 45, 46–48, 50, 51

Odd-Nim, 60Odd-Person-Out, 271

Pascal’s Wager, 160Pick-Up-Bricks, 1, 15, 16, 56Prisoner’s Dilemma, 134, 145, 229, 246Probabilistic Repeated Prisoner’s

Dilemma, 229, 230–232, 234, 236Push, 84

Rock-Paper-Scissors, 89, 90, 173-Dynamite, 129-Lizard-Spock, 106-Superman-Kryptonite, 129Weighted, 129

S-Pick-Up-Bricks, 60, 62SOS, 24

339

340 Index of Games

Split and Choose, 281Stag Hunt, 141, 145, 218, 220, 249

Tic, 4Tragedy of the Commons, 246Triangle Solitaire, 188Turning Turtles, 61Two-Finger Morra, 90, 106

Volunteering Dilemma, 141, 145, 218,220, 249

Voting Scenario, 276

Wythoff’s King, 60Wythoff’s Queen, 62

Index

The bold page numbers in index entries are the pages on whichthe term is defined.

arbitration scheme, 223egalitarian, 240Kalai-Smorodinsky, 240Nash, 224, 224, 226, 228

Arrow’s axioms, 292–293Arrow’s Impossibility Theorem, 292,

290–298Axelrod’s Olympiad, 236–237

balanced Nim position, 48, 49–50best pure response, 143, 163–167best response, 163–167binary expansion, 46, 47, 72, 72Brouwer’s Fixed Point Theorem, 193,

196, 197, 325–327

chance node, 146, 147closure, 73, 86coalition, 251–256, 258–260coalitional form, 253, 255coalitional game, 255, 256, 258–260combinatorial game, 3, 4–6, 9, 26, 301contradiction, proof by, 16convex hull, 124, 125, 217convex set, 124, 125core, 259, 260Cournot Duopoly, 176, 178–181

demand curve, 177

depth, 12, 12, 53domination, 92, 93, 142, 143, 249–250iterated removal, 92–95, 115, 143,168, 250

S-domination, 259strict, 92, 142, 249

dot product, 123, 124dyadic number, 71, 72–74

envy-free division, 280equating results, 113, 114–115, 167, 168equitable division, 278equivalenceclass, 43coalitional game, 272matrix game, 159position, 36, 37–41relation, 37, 43topological, 199, 200

evolutionary stability, 173, 175, 176expected payoff, 98, 99, 135–137, 163expected value, 97, 98extensive form, 151, 154

Fibonacci Sequence, 23Fisher’s Principle, 184fixed point, 192, 193fixed point property, 193, 194–197, 199,

201, 202, 207, 326–328

341

342 Index

Folk Theorem, 229, 234, 234–236

Gale-Shapley algorithm, 287, 287–289game tree, 3–9, 12, 14, 145–154golden ratio, 23, 44, 62guarantee, 100, 100–103guarantee function, 119, 120–122

hyperplane, 123, 124, 131, 318

impartial game, 26, 45–58imputation, 257, 258, 260, 261induction, proof by, 10–12information set, 148, 148–149, 152, 154instant runoff, 276iterated removal of dominated

strategies, 92–95, 115, 143, 168,250

linear programming, 309–313

matrix game, 134, 139–144Matrix-to-Tree Procedure, 153–154MEX (minimal excluded value), 52MEX Principle, 53, 54, 56, 82Minimax Theorem, 313mixed outcome, 214, 217–219move rule, 3movement diagram, 144–145

Nash arbitration, 224, 224, 226, 228Nash equilibrium, 166, 167–169,

187–188Nash Equilibrium Theorem, 166, 167,

187–207, 246Nash flow, 202, 204, 205, 328–330Nash’s axioms, 225negative of a Hackenbush position, 65negotiation set, 219, 220Nim-sum, 50nimber, 50, 51–53node, 4, 5, 7, 145, 146, 149normal form, 151normal-play game, 3, 25–27

outcome, 3, 4–5, 89, 96, 97, 99, 133–139,213–215, 217–220, 245

partition, 43, 44partizan game, 26, 63–83

payoff, 90, 134, 135–137, 139payoff matrix, 162, 163, 215–216payoff polygon, 216–217, 218, 219, 221,

223position, 3, 27, 147–150, 301–308balanced (Nim), 48, 49–50dyadic, 75, 75, 77–79, 81equivalence, 36, 37–41fractional, 69–71integral, 66, 67–68negation, 65sums of, 31terminal, 3type of, 28, 29–30

probability space, 96, 96, 97, 99

random variable, 96

S-domination, 259saddle point, 94, 95, 104security level, 214, 215, 216, 219Shapley Value, 261, 261–267Shapley’s axioms, 263Shapley-Shubik Index, 269, 269simplex, 191, 192, 197, 325Simplicity Principle, 77, 78, 79, 80, 82solution concept, 258solution point, 223Sperner’s Lemma, 189–191, 192,

324–325Sprague-Grundy Theorem, 52–53, 53stable set, 260, 260status quo point, 221, 223stochastic game, 108, 108–109strategic form, 151–153strategyAlternating Trigger, 241dominated, 92, 93, 142, 143, 249–250drawing, 7, 12evolutionarily stable, 175, 176Grim Trigger, 231, 232–233mixed, 98, 99, 101, 102, 162, 163–166pure, 91, 98, 142, 231strictly dominated, 92, 142, 249Tit-for-Tat, 237winning, 7, 12

strategy space, 198, 199–202, 204–207,327–329

Index 343

strategy stealing, 16, 18sum of positions, 31surreal numbers, 307–308symmetric Nash equilibrium, 172, 173,

175symmetry, 15, 16, 65, 172, 225, 264

Tree-to-Matrix Procedure, 151, 152type of a position, 28, 29–30

utility, 135–139utility function, 278

valuation scheme, 261value, 102von Neumann and Morgenstern’s

Lottery, 138–139von Neumann Minimax Theorem, 102,

102, 111, 123–128von Neumann solution, 102, 103,

113–115, 118–122voting game, 268

W-L-D game tree, 5, 6–9, 12–14win rule, 3winning move, 56, 58, 84

Zermelo’s Theorem, 9, 12, 14, 28zero-sum matrix game, 89, 90, 99–102,

111–112, 134

AMS on the Webwww.ams.org

This book offers a gentle introduction to the mathematics of both sides of game theory: combinatorial and classical. The combination allows for a dynamic and rich tour of the subject united by a common theme of strategic reasoning.

The first four chapters develop combi-natorial game theory, beginning with an introduction to game trees and mathematical induction, then investigating the games of Nim and Hackenbush. The analysis of these games concludes with the cornerstones of the Sprague-Grundy Theorem and the Simplicity Principle.

The last eight chapters of the book offer a scenic journey through the math-ematical highlights of classical game theory. This contains a thorough treatment of zero-sum games and the von Neumann Minimax Theorem, as well as a student-friendly development and proof of the Nash Equilibrium Theorem. The Folk Theorem, Arrow’s voting paradox, evolutionary biology, cake cutting, and other engaging auxiliary topics also appear.

The book is designed as a textbook for an undergraduate mathematics class. With ample material and limited dependencies between the chapters, the book is adaptable to a variety of situations and a range of audiences. Instructors, students, and independent readers alike will appreciate the flexibility in content choices as well as the generous sets of exercises at various levels.

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STML/80

For additional information and updates on this book, visit

www.ams.org/bookpages/stml-80